Question

given the figure below, find the values of x and z. (7x + 15)° 74° z°

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So \(7x + 15=z\) and the other pair of vertical angles gives us an equation. Also, since the sum of angles around a point is \(360^{\circ}\), and the non - overlapping angles come in pairs of vertical angles, we know that \(2(7x + 15)+2z = 360\). But we can also use the fact that vertical angles are equal directly. The angle \(7x + 15\) and the angle opposite to it are equal, and the \(74^{\circ}\) angle and its opposite are equal. Since \(7x+15 = z\) and we know that the sum of adjacent angles forming a straight - line is \(180^{\circ}\). We can use the fact that \(7x + 15+74=180\) (because they are supplementary angles).

Step2: Solve for \(x\)

\[

$$\begin{align*} 7x+15 + 74&=180\\ 7x+89&=180\\ 7x&=180 - 89\\ 7x&=91\\ x& = 13 \end{align*}$$

\]

Step3: Solve for \(z\)

Since \(z=7x + 15\), substitute \(x = 13\) into the equation.
\[

$$\begin{align*} z&=7\times13+15\\ z&=91 + 15\\ z&=106 \end{align*}$$

\]

Answer:

\(x = 13\), \(z = 106\)